Consider an infinite chain of point charges plusminus q with

Consider an infinite chain of point charges, plusminus q (with alternating signs), strung out along the x axis, each a distance d from its nearest neighbors. Find the work per particle required to assemble this system. aq^2 The answer will be , for some dimensionless number a. You need to find a. Something helpful is the Taylor series: ln(1 + x) = x - 1/2x^2 +

Solution

Let us try and find potential on origin that is being contributed by individual point charges.

V = kq²/d-kq²/2d+kq²/3d-kq²/4d...

So, V = (kq²/d)*(1-1/2+1/3-1/4+1/5)

now using taylor series ln(2) = 1-1/2+1/3... where x = 1

So V = ln(2)*(kq²/d)

alpha = ln(2)